Solve 2x-1/x-3x^2+1+x/3x^2+x=6(x-1)/9x^2-1

$$\frac { 2 x - 1 } { x - 3 x ^ { 2 } } + \frac { 1 + x } { 3 x ^ { 2 } + x } = \frac { 6 ( x - 1 ) } { 9 x ^ { 2 } - 1 }$$

Solve for x

$$x=1$$

Steps Using Factoring

Variable x cannot be equal to any of the values -\frac{1}{3},0,\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by x\left(3x-1\right)\left(3x+1\right), the least common multiple of x-3x^{2},3x^{2}+x,9x^{2}-1.

$$\left(-1-3x\right)\left(2x-1\right)+\left(3x-1\right)\left(1+x\right)=x\times 6\left(x-1\right)$$

Use the distributive property to multiply -1-3x by 2x-1 and combine like terms.

$$x+1-6x^{2}+\left(3x-1\right)\left(1+x\right)=x\times 6\left(x-1\right)$$

Use the distributive property to multiply 3x-1 by 1+x and combine like terms.

$$x+1-6x^{2}+2x+3x^{2}-1=x\times 6\left(x-1\right)$$

Combine x and 2x to get 3x.

$$3x+1-6x^{2}+3x^{2}-1=x\times 6\left(x-1\right)$$

Combine -6x^{2} and 3x^{2} to get -3x^{2}.

$$3x+1-3x^{2}-1=x\times 6\left(x-1\right)$$

Subtract 1 from 1 to get 0.

$$3x-3x^{2}=x\times 6\left(x-1\right)$$

Use the distributive property to multiply x\times 6 by x-1.

$$3x-3x^{2}=6x^{2}-x\times 6$$

Multiply -1 and 6 to get -6.

$$3x-3x^{2}=6x^{2}-6x$$

Subtract 6x^{2} from both sides.

$$3x-3x^{2}-6x^{2}=-6x$$

Combine -3x^{2} and -6x^{2} to get -9x^{2}.

$$3x-9x^{2}=-6x$$

Add 6x to both sides.

$$3x-9x^{2}+6x=0$$

Combine 3x and 6x to get 9x.

$$9x-9x^{2}=0$$

Factor out x.

$$x\left(9-9x\right)=0$$

To find equation solutions, solve x=0 and 9-9x=0.

$$x=0 x=1$$

Variable x cannot be equal to 0.

$$x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{1}{3},0,\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by x\left(3x-1\right)\left(3x+1\right), the least common multiple of x-3x^{2},3x^{2}+x,9x^{2}-1.

$$\left(-1-3x\right)\left(2x-1\right)+\left(3x-1\right)\left(1+x\right)=x\times 6\left(x-1\right)$$

Use the distributive property to multiply -1-3x by 2x-1 and combine like terms.

$$x+1-6x^{2}+\left(3x-1\right)\left(1+x\right)=x\times 6\left(x-1\right)$$

Use the distributive property to multiply 3x-1 by 1+x and combine like terms.

$$x+1-6x^{2}+2x+3x^{2}-1=x\times 6\left(x-1\right)$$

Combine x and 2x to get 3x.

$$3x+1-6x^{2}+3x^{2}-1=x\times 6\left(x-1\right)$$

Combine -6x^{2} and 3x^{2} to get -3x^{2}.

$$3x+1-3x^{2}-1=x\times 6\left(x-1\right)$$

Subtract 1 from 1 to get 0.

$$3x-3x^{2}=x\times 6\left(x-1\right)$$

Use the distributive property to multiply x\times 6 by x-1.

$$3x-3x^{2}=6x^{2}-x\times 6$$

Multiply -1 and 6 to get -6.

$$3x-3x^{2}=6x^{2}-6x$$

Subtract 6x^{2} from both sides.

$$3x-3x^{2}-6x^{2}=-6x$$

Combine -3x^{2} and -6x^{2} to get -9x^{2}.

$$3x-9x^{2}=-6x$$

Add 6x to both sides.

$$3x-9x^{2}+6x=0$$

Combine 3x and 6x to get 9x.

$$9x-9x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-9x^{2}+9x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 9 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}}}{2\left(-9\right)}$$

Take the square root of 9^{2}.

$$x=\frac{-9±9}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-9±9}{-18}$$

Now solve the equation x=\frac{-9±9}{-18} when ± is plus. Add -9 to 9.

$$x=\frac{0}{-18}$$

Divide 0 by -18.

$$x=0$$

Now solve the equation x=\frac{-9±9}{-18} when ± is minus. Subtract 9 from -9.

$$x=-\frac{18}{-18}$$

Divide -18 by -18.

$$x=1$$

The equation is now solved.

$$x=0 x=1$$

Variable x cannot be equal to 0.

$$x=1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{1}{3},0,\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by x\left(3x-1\right)\left(3x+1\right), the least common multiple of x-3x^{2},3x^{2}+x,9x^{2}-1.

$$\left(-1-3x\right)\left(2x-1\right)+\left(3x-1\right)\left(1+x\right)=x\times 6\left(x-1\right)$$

Use the distributive property to multiply -1-3x by 2x-1 and combine like terms.

$$x+1-6x^{2}+\left(3x-1\right)\left(1+x\right)=x\times 6\left(x-1\right)$$

Use the distributive property to multiply 3x-1 by 1+x and combine like terms.

$$x+1-6x^{2}+2x+3x^{2}-1=x\times 6\left(x-1\right)$$

Combine x and 2x to get 3x.

$$3x+1-6x^{2}+3x^{2}-1=x\times 6\left(x-1\right)$$

Combine -6x^{2} and 3x^{2} to get -3x^{2}.

$$3x+1-3x^{2}-1=x\times 6\left(x-1\right)$$

Subtract 1 from 1 to get 0.

$$3x-3x^{2}=x\times 6\left(x-1\right)$$

Use the distributive property to multiply x\times 6 by x-1.

$$3x-3x^{2}=6x^{2}-x\times 6$$

Multiply -1 and 6 to get -6.

$$3x-3x^{2}=6x^{2}-6x$$

Subtract 6x^{2} from both sides.

$$3x-3x^{2}-6x^{2}=-6x$$

Combine -3x^{2} and -6x^{2} to get -9x^{2}.

$$3x-9x^{2}=-6x$$

Add 6x to both sides.

$$3x-9x^{2}+6x=0$$

Combine 3x and 6x to get 9x.

$$9x-9x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-9x^{2}+9x=0$$

Divide both sides by -9.

$$\frac{-9x^{2}+9x}{-9}=\frac{0}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{9}{-9}x=\frac{0}{-9}$$

Divide 9 by -9.

$$x^{2}-x=\frac{0}{-9}$$

Divide 0 by -9.

$$x^{2}-x=0$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=\frac{1}{4}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{1}{2} x-\frac{1}{2}=-\frac{1}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=1 x=0$$

Variable x cannot be equal to 0.

$$x=1$$